Remark (Reversibility and Ergodicity) 
Each of the two algorithms above are constructed in such a way that they are reversible, which ensures that the target distributions are invariant. As pointed out in [1]}, proving that sampling algorithms on manifolds are ergodic is non-trivial. For [2]}'s algorithm, this is done using the nice properties that are assumed when using a Hamiltonian. For [3]}'s algorithm, ergodicity is a consequence of the additional assumption that their manifolds are compact, connected, and smooth. In the examples in the following section, we apply both algorithms to both compact and non-compact problems.

